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11
votes
1answer
213 views

What can be said about the Fourier transforms of characteristic functions?

What can be said about the Fourier transform of the characteristic function $1_A$, where $A\subset \mathbb{R}^n$ is of finite Lebesgue measure? In particular, What properties are common to ...
2
votes
1answer
181 views

Modulus of of continuity of a convolution operator with respect to Wasserstein metric

For a (discrete) measure $G$ on some reasonable metric space $\Theta$, consider the map $G \mapsto f_G$ defined as $$ f_G := f*G(dx) := \int f(dx|\theta) G(d\theta) $$ for some nice kernel function ...
5
votes
4answers
194 views

Characterizing and counting boolean functions with all influences 1/2

Is there a characterization of boolean functions $f:\{-1,1\}^n \longrightarrow \{-1,1\}$, so that $\mathbf{Inf_i}[f]=\frac{1} {2}$, for all $1\leq i\leq n$? Is it known how many such functions there ...
7
votes
1answer
175 views

Absolute continuity reflected in Fourier coefficients?

Imagine $\mu$ and $\nu$ are two Borel probabilty measures in the interval $[0,1]$. We say that $\mu$ is absolutely continuous with respect to $\nu$, if for every measurable set $A$ such that ...
2
votes
1answer
48 views

Error of Discrete Fourier Transform on Finite Domain (vs. Continuous FT) in terms of Sobolev order

My question is about quantifying the error that occurs by approximating the continuous Fourier transform on a finite domain by using a discretised version with resolution $N$ for a function of a given ...
2
votes
1answer
141 views

Bandwidth approximation for a nonlinear problem

Can anyone please help me with this problem. I must let you know from the beginning that it's not an easy one. "Two functions are given: $u, y \in L^{2}(-\infty,\infty), y(t)=\frac{u(t)}{u(t)+b}$ , ...
2
votes
0answers
69 views

$f, \hat{f} \in L^{p}\cap L^{\infty} \implies f\in B(\mathbb R)$ (algebra of Fourier- Stieltjes transforms )?

For a bounded complex Borel measure $\mu$ on $\mathbb R$, we define, its Fourier-Stieltjes transform, $\hat{\mu}(y)= \int_{\mathbb R} e^{-2\pi ix\cdot y} d\mu(x); (y\in \mathbb R).$ Let $1\leq p \leq ...
2
votes
1answer
167 views

Fourier series of functions on compact groups

Let $G$ be a compact, second countable, Hausdorff topological group with the normalized Haar measure $\mu$. From Peter-Weyl's theorem we now that for any $f\in \mathrm{L}^2(G)$ the Fourier series of ...
29
votes
1answer
573 views

For which maps $S^1\to S^1$ is the winding number defined?

There are two classes of maps $S^1\to S^1$ for which I know how to define the winding number: • Continuous maps: Using the unique path lifting property of the universal covering map $\mathbb R\to ...
0
votes
0answers
52 views

Under condition of Zygmund is the following inequality true?

Let $f:R\rightarrow R$ be a continuous function and satisfies the following Zygmund condition $$ |f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const} \frac{|\delta|}{(\log\frac{1}{|\delta|})^{\alpha}}, ...
1
vote
1answer
56 views

Is the fractional integral of order 1/2 of an L_2 function continuous

Let $R_\alpha f(t) = \int_0^t (t-s)^{-\alpha} f(s)\,ds$ the fractional integration operator. If $f \in L_q(0,1)$ for some $q>2$ then $R_{1/2} f$ is (even Hölder) continuous on $[0,1]$. My ...
0
votes
0answers
46 views

Discrete Fourier tranform on $L^2(\mathbb{R})$

I'm studying on the following construction: For $\lambda\in\mathbb{R}$, denote $$Y_{\lambda}=\left\{g\in L^{2,loc}(\mathbb{R})\,:\,g(t+2\pi)=e^{2\pi i\lambda}g(t)\right\}$$ We give to $Y_{\lambda}$ ...
0
votes
0answers
24 views

properties of frequency-uniform decomposition operator $\square_{k}^{\sigma}$

Let $\rho \in S(\mathbb R^{n})= \text{Schwartz space}, \ \rho:\mathbb R^{n}\to [0,1]$ be smooth radial function verifying $\rho(\xi)=1$ for $|\xi|_{\infty}\leq \frac{1}{2}$ and $\rho(\xi)=0$ for ...
17
votes
1answer
625 views

Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?

Mathematica knows that: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)\;\;\;\;\;\;\;\;\;\;\;\; (1)$$ The von Mangoldt function should then be: ...
0
votes
1answer
152 views

Closed sets in the space of Fourier transforms $\mathcal{F}L^{1}$

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
19
votes
6answers
3k views

explicit formula for Riemann zeros counting function

I've often seen it stated (in vague terms) that there's a Fourier duality between the set of prime numbers and the set of nontrivial Riemann zeta zeros. Because there are various explicit formulae ...
0
votes
1answer
796 views

Sum of product of Fourier series

I want to compute the following product $$\frac{1}{N}\sum_{t=1}^{N}\left(\sum_{s=-\infty}^{\infty}a_{s}\exp(2\pi is\frac{t}{N}\right)\left(\sum_{z=-\infty}^{\infty}a_{z}\exp(-2\pi ...
0
votes
0answers
68 views

when does this identity hold?

When I compute some Fourier series, I find the following interesting identity: $$ \sum_{n\in\mathbb{Z}}\left(\frac{\sin((n+a)x)}{(n+a)x}\right)^k =\frac{1}{|x|}\int_{y\in\mathbb{R}}\left(\frac{\sin ...
2
votes
0answers
39 views

Random square submatrices of a Hadamard matrix

Question: For $N$ be a power of $2$, let $A$ be a random $d \times d$ submatrix of the $N \times N$ Hadamard matrix (the matrix of the Hadamard/Walsh-Fourier transform). What is the best known upper ...
1
vote
1answer
235 views

Is this set a Riesz Basis of $L^2(0,\pi)$

Let $A=\{\sqrt{2}\sin(\sqrt{n^2+a} \pi x)\} _{n=1}^\infty$, where $a$ is a positive real number. Is $A$ a Riesz Basis of $L^2(0,1)$?
2
votes
2answers
258 views

A sufficient condition for a probability measure to have compact support

Consider a probability measure $\mu$ on, let's say, $\mathbb R$. Is there a necessary and sufficient condition so that $\mu$ has compact support $Supp(\mu)$ ? I agree this question is too vague, ...
1
vote
1answer
86 views

norm of trigonometric polynomials under arbitrary change of signs

Let $\mathcal P$ be the collection of trigonometric polynomial with $L^p[0,1]$ norm one, $p>2$, then by changing the sign of coefficients of polynomials in $\mathcal P$ randomly, can we produce ...
14
votes
2answers
379 views

What geometric information is carried by the Fourier coefficients of the components of a closed curve?

Let $\gamma$ be a smooth closed curve in the plane and let $(x(t), y(t))$ be a parametrization. The functions $x(t)$ and $y(t)$ are smooth and periodic, so each has a uniformly convergent Fourier ...
1
vote
1answer
83 views

Absolute convergence of multi-dimensional Fourier series

For a Lipschitz function $f$ defined in $[0,2\pi]^d$ for $d>1$, is that true that the multi-dimensional Fourier series converges absolutely? In other words, $\sum_{k\in ...
0
votes
1answer
114 views

Is $\int_0^\infty \sin(Kx)f_K(x)\,dx$ of larger order than $\int_0^\infty \cos(Kx)f_K(x)\,dx$?

Suppose we have a function $f$, such that $f$ is of some smoothness degree $m$, and $f,f^{(k)} \in L_1[0,\infty)$ $k=1,...,m$. Now if $f^{(k)}(0) = \lim_{x\rightarrow\infty}f^{(k)}(x) = 0$ for ...
3
votes
0answers
91 views

$f,g , |f|f, |g|g \in A(\mathbb R) \ \text{(Banach algebra)} \implies \left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$?

Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra $$A(\mathbb R):= \{f\in ...
0
votes
1answer
147 views

Can we expect $\left\||f|^{2}f-|g|^{2}g\right\|\leq C ||f-g||$ in the Banach algebra $A(\mathbb T)$ ?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb T):= \{f\in ...
0
votes
0answers
73 views

Bounding the norm of the Dirichlet kernel as a matrix function

I asked the following there also, but I am beginning to think it would be more appropriate here. Consider the Dirichlet kerel: $f(x) = 1+2\sum_{k=1}^{N}\cos(kx)$. Now, given a diagonalizable real ...
5
votes
1answer
192 views

Fourier expansion of Takagi-function (everywhere non differentiable function).

Let us consider Takagi-function defined by $T(x) \colon\!= \sum_{n=0}^{\infty}s(2^nx)/2^n$, where $s(x) \colon\!\!= \underset{n \in {\Bbb Z}}{\mathrm{min}} \,|x-n|$. $T(x)$ has its period $1$, so ...
1
vote
2answers
80 views

Numerical calculation of Fourier transform with a nice error bound

I'd like to have an algorithm for a numerical calculation of Fourier transform with a nice error bound. To be precise, if $f$ is a function from $L_1(R)$, $F[f]$ is it's exact Fourier transform and ...
2
votes
0answers
70 views

Positive Fourier coefficients for a function $f:\{+1,-1\}^n \to \mathbb R$

This is from my research in computer science where the Fourier transform over $GF(2)^n$ is a tool to study functions on the Boolean hypercube. For example, the majority function on 3 variables is ...
1
vote
0answers
86 views

How to use, $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in a Banach space $C([0,T]; M^{p,1})$?

(May be this is very basic question for MO) (For details or this question you may see the paper page no. 9, MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; ...
7
votes
2answers
421 views

Are Fourier series of length 2 'asymmetric enough' to generate all crossing patterns? - A reformulation of the Fourier-(1,1,2) knot question

Given $N$ pairs of distinct real numbers $t_i, t'_i \in [0,1]$, $i = 1,\ldots,N$, we ask if there is a function $f(x) = \cos(2\pi mx+\alpha) + \gamma\cdot \cos(2\pi nx+\beta)$, with $m, n \in ...
2
votes
1answer
104 views

Fourier inversion

I have certain doubts about a classical Fourier inversion theorem. According to it (this is a theorem from "Panorama of Harmonic Analysis" by Krantz), if $f$ and $\hat{f}$ are both in $L_1(R)$ and ...
0
votes
0answers
56 views

Fourier Analysis in Kahane and Zelasko's characterization of maximal ideals in commutative Banach algebra

I've been told that Kahane and Zelasko (in A characterization of maximal ideals in commutative Banach algebra's Studia Math 29 1968) use a "non-trivial result" from Fourier analysis in their proof of ...
1
vote
0answers
53 views

Counting frequencies of occurrence of patterns within a sequence using harmonic analysis?

Assume that we are given a sequence $\mathbf x := X_1,\dots,X_n \in \mathbb N^n$ for some $n \in \mathbb N$. I am interested in calculating the frequency of occurrence of some fixed sequence $\mathbf ...
0
votes
0answers
88 views

Localized convolution operators

Take our favorite differential operator $\Delta$ on $\mathbb{R}^3$. Module a constant, we can represent its inverse (in some sense) by a convolution operator, namely $$\Delta^{-1}f=f*\frac{1}{|x|}.$$ ...
3
votes
3answers
385 views

Modular form on $\Gamma_0(N)$

I recently asked this question on Math.StackExchange with no answer so far. So I thought maybe I can find an answer here. Let $M(k,\Gamma_0(N))$ be a space of modular forms of weight $k$ on ...
9
votes
1answer
201 views

Determinant of $V^* V$ where $V$ is rectangular Vandermonde matrix with nodes on unit circle

Let $z_{1},\dots,z_{k}$ be distinct complex numbers with $\left|z_{j}\right|=1,\;j=1,\dots,k$. For any natural $N\geqslant k$ consider the rectangular Vandermonde matrix $$ V_{N}=\begin{pmatrix}1 ...
2
votes
2answers
100 views

Seeking a class of functions for which sums approximate integrals well

Is there a "natural" class of integrable functions $f: {\mathbb R} \rightarrow {\mathbb R}$ for which it is true (and, preferably, not too hard to prove!) that $\sup_{0 \leq a < h} |h S(a,h) - I|$ ...
1
vote
1answer
79 views

Carleson-Hunt inequality: changing order of summation

The well-known Carleson-Hunt inequality states that for any $p>1$ and any $f \in L^p(0,2\pi)$, the $L^p$-norm of $sup_n |S_n|$, where $S_n$ is the n-th partial sum of the Fourier series of $f$, is ...
4
votes
0answers
85 views

Error of midpoint method for differentiable functions

Is it the case that for every differentiable function $f$ on $[0,1]$ (with finite one-sided derivatives at the endpoints), the midpoint method of estimating $\int_0^1 f(x) \: dx$ has error $o(1/n)$? ...
0
votes
0answers
45 views

How to choose negative definite function $\lambda (x)$, so that $\lambda^{-1} \in L^{1}(\mathbb R)$?

We define, $$\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha), (\mu(0)=0) $$ where $\mu (\alpha)$ is a non-decreasing function such that the integral converges for ...
-2
votes
1answer
139 views

$L^{1}(\mathbb R) \cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R)\subset H_{1}(\mathbb R)$?

Put, $C_{0} (\mathbb R)=\{f:\mathbb R \to \mathbb C: f \text { is continuous on} \ \mathbb R \ \text {and } \lim_{|x|\to \pm \infty}f(x)=0 \}$(= Continuous functions on $\mathbb R$ vanishing at ...
1
vote
1answer
119 views

Result of Beurling concerning absolute convergence of Fourier series of |f|

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$ and we put, $$A(\mathbb T):= \{f\in ...
1
vote
0answers
71 views

Relationship between Fourier series & DFT

Sources like http://www.dsprelated.com/dspbooks/mdft/Relation_DFT_Fourier_Series.html explain the equivalence between FS and DFT. However, isn't there a flaw? When I integrate over the continuous ...
7
votes
0answers
159 views

Laplace Transform in the context of Gelfand/Pontryagin

Question: Do quasi-characters properly generalize the Laplace transform or decompose functions in some setting in a way similar to how characters generalize the Fourier transform and decompose $L^1$ ...
1
vote
1answer
222 views

When one can expect $\widehat{(fg)} = \hat{f} \ast \hat{g}$; $f, g\in L^{1} (G)$?

Let $f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of $f$ as follows: $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$ It is ...
3
votes
1answer
125 views

What does a Zonal sphere harmonic look like?

Let $\mathcal{H}_k\subset L^2(S^n)$ be the space of sphere harmonics of degree $k$, i.e., they are the eigenfunctions of $\Delta_{S^n}$. And let $\Pi_k:L^2(S^n)\to \mathcal{H}_k$ be the orthogonal ...
0
votes
0answers
49 views

How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)?

We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and ...